Predicate Calculus Control Strategies for Resolution Methods

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Breadth-First StrategyAll of the first-level resolvents are computed first, then the second-level resolvents,and so on. A first-level resolvent is one between two clauses in the base set; an i-thlevel resolvent is one whose deepest parent is an (i-1)-th level resolvent. The breadthfirststrategy is complete, but it is grossly inefficient.Example: Find all resolvents for the following base set: ¬I(z) ∨ R(z), I(a),¬R(x)∨ L(x), ¬D(y) ∨ ¬L(y), D(a)The Set-of-Support StrategyAt least one parent of each resolvent is selected from among the clauses resultingfrom the negation of the goal or from their descendants. The strategy is complete.The Unit-Preference StrategyIt is a modification of the set-of support strategy in which, instead of filling out eachlevel in breadth-first fashion, we try to select a single-literal clause (called a unit) tobe a parent in a resolution. Every time units are used in resolution, the resolvent havefewer literals than do their other parents. The strategy is complete.The Linear-Input Form StrategyEach resolvent has at least one parent belonging to the base set. The strategy is notcomplete.The Ancestry-Filtered Form StrategyEach resolvent has a parent that is either in the base set or that is an ancestor of theother parent. The strategy is complete.Example: Find all resolvents for the following base set: ¬Q(x) ∨ ¬P(x), ¬Q(y) ∨¬P(y), ¬Q(w) ∨ P(w), Q(u) ∨ P(a).Problem 1.Translate into symbols the following statements, using quantifiers, variablesand predicate symbols.If some trains are late then all trains are late.Everyone is loyal to someoneSome people hate everyone.No mouse is heavier than any elephant.Problem 2.Consider the following statements:- if the maid stole the jewelry, then the butler wasn’t guilty.- either the maid stole the jewelry or she milked the cows.- if the maid milked the cows, then the butler got his cream.- therefore, if the butler was guity, then he got his cream.a) Express these statements in the propositional calculus.
) Express the negation of the conclusion in clause form.c) Demonstrate that the conclusion is valid, using resolution in thepropositional calculus.Problem 3.Check if the formula (A -> ~A) -> (A ∧ (~A -> A)) is a tautology.Problem 4.Find CNF for the following formula:(∀x)(∃y)[ Q(x,y) ∧ ¬P(x,y)] ∨ (∃y) (∀x )[Q(x,y) ∧ ¬R(x,y)].Problem 5.Translate into symbols the following statements, using quantifiers, variablesand predicate symbols:- Tony, Mike, and John belong to the Alpine club.- Every member of the Alpine club who is not a skier is a mountain climber.- Mountain climbers do not like rain and anyone who does not like snow isnot a skier.- Mike dislikes whatever Tony likes and likes whatever Tony dislikes.- Tony likes rain and snow.Use resolution to show that:- There a member of the Alpine club who is a mountain climber but not askierProblem 6.Translate into symbols the following statements:- If a course is easy, some students are happy.- If a course has a final, no students are happy.Use resolution to show that: If a course has a final, the course is not easy.

Predicate Calculus Control Strategies for Resolution Methods

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